Random walks on directed percolation clusters

2 downloads 0 Views 287KB Size Report
Mar 13, 2016 - Random walks on directed percolation clusters. View the table of contents for this issue, or go to the journal homepage for more. 1982 J. Phys.
Home

Search

Collections

Journals

About

Contact us

My IOPscience

Random walks on directed percolation clusters

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1982 J. Phys. A: Math. Gen. 15 L189 (http://iopscience.iop.org/0305-4470/15/4/007) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 130.225.169.1 This content was downloaded on 13/03/2016 at 15:55

Please note that terms and conditions apply.

J. Phys. A: Math. Gen. 15 (1982) L189-L193. Printed in Great Britain

LElTER TO THE EDITOR

Random walks on directed percolation clusters T Vicsek, J KertCszt and J Cserti Research Institute for Technical Physics, Budapest, Pf 76,1325 Hungary Received 14 December 1981

Abstract. An asymptotically exact expression is given for the mean displacement (R(r)) of random walks on directed percolation clusters on lattices in arbitrary dimensions. The critical behaviour of R, = lim,+,(R(t)), the mean squared displacementand the relaxation time is discussed near the threshold probability pc = 1 in terms of critical exponents.

Many versions of the percolation problem have been studied since the classical work of Broadbent and Hammersley (1957). Recently more attention has been focused on developing sophisticated models which may meet theoretical interest as well as possible applications in various fields (for references see Stauffer (1979) and Essam (1980)). In the case of directed percolation the bonds transmit only in one direction. This direction can be specified in a number of ways: the most common model is obtained when a bond is directed like a unit vector parallel to it and having positive projection on a fixed vector, playing the role of an external field. Thus on the square lattices the bonds are directed downward or to the right (see figure 1). This model is in a different universality class from ordinary percolation (Blease 1977a, b, KertCsz and Vicsek 1980, Obukhov 1980, Kinzel and Yeomans 1981) and has relevance to

Figure 1. Part of the directed square lattice. All bonds are directed like bonds al and a2. The distance travelled by a particle starting from the origin and arriving at the point mo after k = ko steps is marked by a broken line.

t Present address: Institute for Theoretical Physics, University of Koln, D-5000 Koln 41, West Germany.

0305-4470/82/040189 + 05$02.00 @ 1982 The Institute of Physics

L189

L190

Letter to the Editor

phenomena occurring in various systems, such as hopping conduction in a semiconductor (KertCsz and Vicsek 1980, Van Lien and Shklovskii 1981), Markov process with branching, chemical processes, absorption and recombination (Grassberger and de la Torre 1979). Random walks on ordinary percolation clusters have been investigated mainly in connection with diffusion in disordered lattices. Since the introduction of the problem called 'the ant in the labyrinth' (de Gennes 1976), a few papers have dealt with the ) the ant as a function of time. description of the mean squared displacement ( R 2 ( t )of The main quantities of the problem, the critical exponent for the long-time limit of ( R 2 ( t ) )and the associated relaxation time were obtained for site (Roussenq 1980) and bond (Vicsek 1981) percolation problems by Monte Carlo and in a closed form for the Bethe lattice (Straley 1980) and for one-dimensional bond percolation (Odagaki and Lax 1980). A natural combination of the above models is a process defined by random walks on directed percolation clusters, which may serve as a model for the electron hopping in a strong external field, when hops opposite to the field are realised with a relatively small probability (KertCsz and Vicsek 1980, Van Lien and Shklovskii 1981). In this letter we investigate the case when no hops are allowed in the direction opposite to the external field; of course this model is an extreme version of the process occurring in a real system. There is a correspondence between random walks on undirected percolation networks and the electrical conductivity of random resistor networks, since the diffusion constant defined from the mean squared displacement of a particle can be related to the conductivity of a resistor network (Mitescu and Roussenq 1976). This analogy, however, does not exist in the case of directed models: the conductivity of a random network of diodes cannot be obtained from the study of random walks; moreover-as will be shown later-even the critical points of the two models are different. We define the random walk of a particle on the directed lattice in the following way. The particle starts moving from an arbitrary site along the directed bonds occupied with probability p. It is allowed to make a directed move in every unit time. The bond through which the step is made is randomly chosen from the occupied ones leading away from the given site. If there are no such bonds the particle stops moving. Let us consider a lattice of coordination number z = 21 with I directed bonds leading to and out from every site ( I = 2 for the square, I = 3 for the triangular and the simple cubic lattices). Furthermore we call this an I lattice. With probability (1- p ) ' the particle having arrived at a site has no opportunity to proceed: it continues walking with probability 1- (1-PI'. The probability of completing k consecutive steps and then being blocked is (1- p ) ' [ 1- (1- p)'Ik. Therefore the mean distance of the particle from the origin after t steps ( t plays the role of time) can be obtained from

c g(k)rl-(l-p)'lk+g(t)rl-(l-p)'l"' I

(R(t))=(l-p)'

(1)

k=O

where in the case of the square lattice k

g ( k )= k

C= O ( k ! / m ! ( -m)!)[1-2m/k k

+ 2 m 2 / k 2 ] " 2 / 2 k= kf(k)

(2)

m

denotes the mean distance of the k-step walk. In (1) the first sum is associated with

Letter to the Editor

L191

the walks stopped not later than in the tth step, while the second term corresponds to the walks which go farther than the tth step. In deriving (2), we considered that a particle initially being at point 0 (figure 1)can reach the point mo in the koth layer in ko!/mo!(ko-mo)!ways. It can be easily shown that f(k)is a slowly varying function and converges for k + m to 1/hin the case of the square lattice. For example f(1) = 1.0, f(l0) = 0.740 and f(50)= 0.714 (1/h= 0.707), showing that the average trajectory of the particle is near the symmetry axis. Since we are interested in the long-time divergence of (R( t ) ) , in the following we use

f(k)=F = k-rm lim f(k).

(3)

It can be mentioned that with a properly chosen non-Euclidean metric the factor f(k) can be made unity (on the square lattice this is IxI+\yl instead of ( ~ ' + y ' ) ' ' ~ for the distance). Introducing w = 1-(1 -p)', we obtain from (1)

i

(R( t ) )= (1- w ) k =O g(k)w

+ g ( t ) w r + l --F-(1l - w W

- w ')

(4)

where only approximation (3) has been utilised. The simple formula (4) contains all the information we need to describe the critical behaviour of the walk. The quantities of interest are the long-time limit of (R (t)) (5)

R m -Fw/(l- W ) and the relaxation time 7, defined by ( R ( t ) ) = R m ( l - ~e-'"). From equations (4)-(6) we obtain 7

-1

=ln(l/w)

and Q = 1. ( R ( t ) )vanishes in isotropic percolation; thus ( R Z ( t )is) used there. This quantity can also be calculated for the directed model:

(R'(t))=F'[w(l+ w ) / ( l - w)'][l-

(1- w)l

(8) with a constant F'. As far as the exponents of R and T are concerned, (8) does not contain new information compared with (4), but the time dependence of the prefactors is changed. Next we are interested in the critical behaviour of R m and 7. They diverge only when w + 1; therefore the critical point of the problem is pc = 1, which result also shows the one-dimensional character of the directed walks. From (5)-(7) one can see that both R m and 7 diverge with the same exponent equal to unity if (1- w ) is taken as scaling parameter. w t - d - l

R,=(I-w)-',

(9Q)

w)-'.

(9b1

7oC(1-

Although this result has been derived for bond percolation on I lattices of arbitrary dimensional lattices, it can be shown that (9) is universally valid for site percolation and for arbitrary lattices too, if w is properly chosen. But, if Ap = 1- p is introduced as a scaling field the exponents are no more universal as it can be demonstrated for the

L192

letter to the Editor

case of an 1 lattice:

R,a (Ap)-',

(loa)

(106) In the problem of one-dimensional, long-range percolation (Klein er a1 1978) and the one-dimensional continuum percolation (Shalitin 198l ) , a similar situation occurs: universality is valid only if appropriate scaling parameters are chosen. In conclusion, we obtained an explicit expression for the mean distance for the problem of random walks on directed lattices. The critical point of this problem is p c = 1, therefore no infinite walk occurs in the system, unless all the bonds are occupied. Above the threshold of directed percolation (-0.64 in the square bond problem, see e.g. Kinzel and Yeomans (1981)) there are infinite directed paths, but if p < 1 the random walker will be caught with probability one in one of the dead branches. The directed walks belong to one universality class irrespective of the type and dimensionality of the lattice, if the scaling field is properly chosen. This universality class is characterised by the one-dimensional directed walk. (Note, the exponent of T is different in the problem of walks on one-dimensional undirected percolation clusters (Odagaki and Lax 1980).) An interesting generalisation of the model is the problem of random walks on a lattice in which steps against the direction of a bond are also allowed with a certain probability q. An unusual crossover is expected in this case since for any values of 4 different from zero the critical concentration of bonds is equal to the undirected bond percolation threshold probability of the given lattice, and the p c ( q = 0) = 1value 'jumps' to the new p c ( 4 # 0) value discontinuously as a function of q. Work along these lines is in progress. The pc = 1 value is also different from the critical point of a corresponding random network of diodes (Redner 1981) because a current through the diodes sets in as soon as a directed path appears in the network, but a randomly walking particle finds this path with zero probability. The diff usion-conduction analogy breaks down when nonlinear elements are introduced into the model, as expected. T a (Ap)-'.

The authors are grateful to T Geszti for many helpful discussions and to D Stauffer for a critical reading of the manuscript. Thanks are due to the financial support by SFB 125 Aachen-Jiilich-Koln extended to one of us (JK). Noteadded. After submittingour letter a closely related letter (Stephen 1981) appeared, in whichageneralised model is studied within the framework of the effective medium approximation. Part of its results disagree with ours. Stephen M J 1981 J. Phys. C: Solid State Phys. 14 L1077

References Blease J 1977a J. Phys. C: Solid State Phys. 10 917 -1977b J. Phys. C: Solid State Phys. 10 925 Broadbent S R and Hammersley J M 1957 Roc. Camb. Phil. Soc. 53 629 Essam J W 1980 Rep. Prog. Phys. 43 833 de Gennes P G 1976 La Recherche 7 916 Grassberger P and de la Torre A 1979 Ann. Phys. 122 373

Letter to the Editor Kertbz J and Vicsek T 1980 J. Phys. C: Solid State Phys. 13 L343 Kinzel W and Yeomans J M 1981 J. Phys. A : Math. Gen. 14 L163 Klein W,Stanley H E, Redner S and Reynolds P J 1978 J. Phys. A : Math. Gen. 11 L17 Mitescu C and Roussenq J 1976 C.R. Acud. Sci., Pans 283 A 999 Obukhov S P 1980 Physica A 101 145 Odagaki T and Lax M 1980 Phys. Rev. Lett. 45 847 Redner S 1981 J. Phys. A : Math. Gen. 14 L349 Roussenq J 1980 Thesis UniversitC de Provence Shalitin D 1981 J. Phys. A : Math. Gen. 14 1983 Stauffer D 1979 Phys. Rep. 54 1 Straley J P 1980 J. Phys. C: Solid State Phys. 13 2991 Van Lien N and Shklovskii B I 1981 Solid State Commun. 38 99 Vicsek T 1981 2.Phys. B 45 153

L193